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Is there any test for whether or not the regression function is even?

Suppose we have a model: $Y=g(X, \epsilon)$, where $Y, X$ are both one dimensional. My questions is how do we test for $g$ is an even function or not? For example $Y=X^2+\epsilon$ (here $g(x)=x^2$, which is even on $x$).

Thanks.

bankrip
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  • I think we're going to need more context here. What is "the regression function"? – hmakholm left over Monica Oct 29 '14 at 16:26
  • By regression function, I mean $E(Y|X)$ – bankrip Oct 29 '14 at 17:06
  • This isn't exactly what you asked for, but a very elementary approach to a similar problem is this: you can use an F-test on the residuals of $\beta_2 X^2 + \beta_1 X + \beta_0$ vs. the residuals of $\beta_1' X_1 + \beta_0'$. In other words, you can test to determine whether or not a quadratic model explains the variability better than a linear model. – Nicolás Kim Oct 29 '14 at 17:39

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